Theorem imass1 5840

Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.)

Assertion

Ref	Expression
imass1	⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶))

Proof of Theorem imass1

Step	Hyp	Ref	Expression
1		ssres 5761	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶))
2		rnss 5691	. . 3 ⊢ ((𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶) → ran (𝐴 ↾ 𝐶) ⊆ ran (𝐵 ↾ 𝐶))
3	1, 2	syl 17	. 2 ⊢ (𝐴 ⊆ 𝐵 → ran (𝐴 ↾ 𝐶) ⊆ ran (𝐵 ↾ 𝐶))
4		df-ima 5456	. 2 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
5		df-ima 5456	. 2 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶)
6	3, 4, 5	3sstr4g 3933	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶))

Syntax hints:   → wi 4   ⊆ wss 3859  ran crn 5444   ↾ cres 5445   “ cima 5446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-cnv 5451  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456

This theorem is referenced by:  vdwnnlem1  16160  dprdres  18867  imasnopn  21982  imasncld  21983  imasncls  21984  utoptop  22526  restutop  22529  ustuqtop3  22535  utopreg  22544  metustbl  22859  imadifxp  30041  esum2d  30969  eulerpartlemmf  31250  brtrclfv2  39576  frege97d  39601  frege109d  39606  frege131d  39613  hess  39630  resimass  41070  setrecsss  44303